# Training-free Spatially Grounded Geometric Shape Encoding (Technical Report)

arXiv: https://arxiv.org/abs/2604.07522 · HTML: https://mlsanity.com/paper/2604.07522/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 30 of 30)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [Make-A-Scene: Scene-Based Text-to-Image Generation with Human Priors](https://mlsanity.com/paper/2203.13131/) (2022)
- [SDR - half-baked or well done?](https://mlsanity.com/paper/1811.02508/) (2018)
- [High Frequency Component Helps Explain the Generalization of Convolutional Neural Networks](https://mlsanity.com/paper/1905.13545/) (2019)
- [Octree Generating Networks: Efficient Convolutional Architectures for High-resolution 3D Outputs](https://mlsanity.com/paper/1703.09438/) (2017)
- [ReCo: Region-Controlled Text-to-Image Generation](https://mlsanity.com/paper/2211.15518/) (2022)
- [Pyroomacoustics: A Python package for audio room simulations and array processing algorithms](https://mlsanity.com/paper/1710.04196/) (2017)
- [Multi-Scale Representation Learning for Spatial Feature Distributions using Grid Cells](https://mlsanity.com/paper/2003.00824/) (2020)
- [Fourier Position Embedding: Enhancing Attention's Periodic Extension for Length Generalization](https://mlsanity.com/paper/2412.17739/) (2024)
- [FutureMapping: The Computational Structure of Spatial AI Systems](https://mlsanity.com/paper/1803.11288/) (2018)
- [AutoGeo: Automating Geometric Image Dataset Creation for Enhanced Geometry Understanding](https://mlsanity.com/paper/2409.09039/) (2024)
- [Towards General-Purpose Representation Learning of Polygonal Geometries](https://mlsanity.com/paper/2209.15458/) (2022)
- [3DPPE: 3D Point Positional Encoding for Multi-Camera 3D Object Detection Transformers](https://mlsanity.com/paper/2211.14710/) (2022)
- [ReZero: Region-customizable Sound Extraction](https://mlsanity.com/paper/2308.16892/) (2023)
- [PolygonGNN: Representation Learning for Polygonal Geometries with Heterogeneous Visibility Graph](https://mlsanity.com/paper/2407.00742/) (2024)
- [Deep Learning for Classification Tasks on Geospatial Vector Polygons](https://mlsanity.com/paper/1806.03857/) (2018)
- [ShapeEmbed: a self-supervised learning framework for 2D contour quantification](https://mlsanity.com/paper/2507.01009/) (2025)
- [Unit Region Encoding: A Unified and Compact Geometry-aware Representation for Floorplan Applications](https://mlsanity.com/paper/2501.11097/) (2025)

## Built on by (most cited first, 1 of 1)

- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)

## Abstract (arXiv)

Positional encoding has become the de facto standard for grounding deep neural networks on discrete point-wise positions, and it has achieved remarkable success in tasks where the input can be represented as a one-dimensional sequence. However, extending this concept to 2D spatial geometric shapes demands carefully designed encoding strategies that account not only for shape geometry and pose, but also for compatibility with neural network learning. In this work, we address these challenges by introducing a training-free, general-purpose encoding strategy, dubbed XShapeEnc, that encodes an arbitrary spatially grounded 2D geometric shape into a compact representation exhibiting five favorable properties, including invertibility, adaptivity, and frequency richness. Specifically, a 2D spatially grounded geometric shape is decomposed into its normalized geometry within the unit disk and its pose vector, where the pose is further transformed into a harmonic pose field that also lies within the unit disk. A set of orthogonal Zernike bases is constructed to encode shape geometry and pose either independently or jointly, followed by a frequency-propagation operation to introduce high-frequency content into the encoding. We demonstrate the theoretical validity, efficiency, discriminability, and applicability of XShapeEnc via extensive analysis and experiments across a wide range of shape-aware tasks and our self-curated XShapeCorpus. We envision XShapeEnc as a foundational tool for research that goes beyond one-dimensional sequential data toward frontier 2D spatial intelligence.